On numerical evaluation of the Heun functions

Motygin, Oleg V.

doi:10.1109/dd.2015.7354864

Cited by 18 publications

(17 citation statements)

References 11 publications

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“…We also note that in view of (16), if Re(−εz) > 0, it may reasonable to compute c Hl (q, α, γ, δ, ε; z) through c Hl (q − εγ, α − ε(γ + δ), γ, δ, −ε; z); see (11), (12). This trick is used in the code [17].…”

Section: Basic Algorithmmentioning

confidence: 99%

“…The purpose of the present work is to develop alternative algorithms. Following [16], for numerical evaluation of the confluent Heun functions we suggest a procedure based on power series, asymptotic expansions and analytic continuation. Program realization is presented in [17] as Octave/Matlab code.…”

Section: Introductionmentioning

confidence: 99%

“…When γ is a nonpositive integer, one solution of (1) is analytic in a vicinity of z = 0 but it is equal to zero at z = 0, whereas the second solution can be normalized to unity at zero but generally it is not analytic. Following [16], the normalized solution will be denoted by c Hl (z) and another one by c Hs(z).…”

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confidence: 99%

“…where c H A,∞ (q, α, γ, δ, ε; z) is the function defined by (15) and E ± 1 , E ± 2 are some constants. It is important that the function in the left-hand side of (25) does not contain exponential factor while each of the functions in the right-hand side does generally contain (via contribution of c H B,∞ (z); see (16)). So it is reasonable to choose the matching point to be close to zero and continue c H A,∞ (z) from far-field to this point (not vice versa).…”

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confidence: 99%

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On evaluation of the confluent Heun functions

Motygin

2018

2018 Days on Diffraction (DD)

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In this paper we consider the confluent Heun equation, which is a linear differential equation of second order with three singular points -two of them are regular and the third one is irregular of rank 1. The purpose of the work is to propose a procedure for numerical evaluation of the equation's solutions (confluent Heun functions). A scheme based on power series, asymptotic expansions and analytic continuation is described. Results of numerical tests are given. arXiv:1804.01007v1 [math.NA] 2 Apr 2018 2 Statement and basic notationsWe use the following form of the confluent Heun equation:This second order linear differential equation has regular singularities at z = 0 and 1, and an irregular singularity of rank 1 at z = ∞ (see e.g. [20]). The parameter q ∈ C is usually referred to as an accessory or auxiliary parameter and γ, δ, ε, α (also belonging to C) are exponent-related parameters.It is important to note that in this paper the parameters ε and α are assumed to be independent. Below, we will use notation c H (q, α, γ, δ, ε; z) or c H (z) for brevity. There are 6 local solutions of equation (1). The Frobenius method can be used to derive local power-series solutions to (1) near z = 0 and z = 1 (two per a singular point), while two solutions at z = ∞ can be obtained in the form of asymptotic series. In § 3 we will present the local solutions near the point z = 0. One of them is analytic in a vicinity of zero and if γ is not a nonpositive integer, we normalize this solution to unity at z = 0 and call it the local confluent Heun function. It is denoted by c Hl (q, α, γ, δ, ε; z). For the second Frobenius local solution, we will use the notation c Hs(q, α, γ, δ, ε; z). When γ is a nonpositive integer, one solution of (1) is analytic in a vicinity of z = 0 but it is equal to zero at z = 0, whereas the second solution can be normalized to unity at zero but generally it is not analytic. Following [16], the normalized solution will be denoted by c Hl (z) and another one by c Hs(z).

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Section: Basic Algorithmmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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On evaluation of the confluent Heun functions

Motygin

2018

2018 Days on Diffraction (DD)

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“…A recent work of Oleg V. Motygin gives alternative algorithms for the numerical evaluation of the Heun function [24].…”

Section: Heun Equationmentioning

confidence: 99%

Quantum field theory applications of heun type functions

Birkandan

Hortaçsu

2017

Reports on Mathematical Physics

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After a brief introduction to Heun type functions we note that the actual solutions of the eigenvalue equation emerging in the calculation of the one loop contribution to QCD from the Belavin-Polyakov-Schwarz-Tyupkin instanton and the similar calculation for a Dirac particle coupled to a scalar CP 1 model in two dimensions can be given in terms of confluent Heun equation in their original forms. These equations were previously modified to be solved by more elementary functions. We also show that polynomial solutions with discrete eigenvalues are impossible to find in the unmodified equations.

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Symbolic and numerical analysis in general relativity with open source computer algebra systems

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We study three computer algebra systems, namely SageMath (with SageManifolds package), Maxima (with ctensor package) and Python language (with GraviPy module), which allow tensor manipulation for general relativity calculations along with general algebraic calculations. We present a benchmark of these systems using simple examples. After the general analysis, we focus on the SageMath and SageManifolds system to derive, analyze and visualize the solutions of the massless Klein-Gordon equation and geodesic motion with Hamilton-Jacobi formalism. We compare our numerical result of the Klein-Gordon equation with the asymptotic form of the analytical solution to see that they agree. * Man = Manifold (4 , 'Man ' , r '\ mathcal { M } ') 4 5 # Define the parameter " M " ( mass ) : 6 M = var ( 'M ') 7 8 # Define the coordinates { t =0 , r =1 , theta (= th ) =2 , phi =3} with ranges 9 # ( BL = Boyer -Lidquist ) 10 BL . = Man . chart (r ' t r :(0 ,+ oo ) th :(0 , pi ) :\ theta ph :(0 ,2* pi ) :\ phi ') 11 12 # Define the metric " g " on manifold " Man ": 13 g = Man . lorentzian_metric ( 'g ') 14 15 # Enter the metric components : 16 g [0 ,0] = (1 -(2* M ) / r ) 17 g [1 ,1] = -1/(1 -(2* M ) / r ) 18 g [2 ,2] = -r^2 19 g [3 ,3] = -( r * sin ( th ) )^2 20 21 # Einstein tensor 40 ET = g . ricci () -(1/2) * g * g . ricci_scalar () 41 # Display all components 42 ET . set_name ( 'E ') 43 show ( ET . display () ) 44 # Display a single component 45 show ( ET [1 ,2]) ¦ ¥ The lines between 16-19 defines the Schwarzschild metric. In order to define the Kerr metric, we need to change this part with § ¤ 16 a = var ( 'a ') 17 rhosq = r^2+( a^2) * cos ( th )^2 18 Delta = r^2 -2* M * r + a^2 19 20 g [0 ,0] = (1 -(2* M * r ) / rhosq ) 21 g [1 ,1] = -rhosq / Delta 22 56 # We can analyze the Klein -Gordon equation 57 # to see how it is separated into 58 # radial and angular parts . 59 60 # Common factors will be divided 61 # This part should be given by the user 62 divideKGby = exp ( -I * omega * t ) * exp ( I * k * ph ) * sin ( th ) * R * S 63 finalKG = expand ( KG / divideKGby ) 64 65 # Extract the operands in the expression : 66 fkgops = finalKG . operands () 67 68 # Find radial and angular parts : 69 # lambda_aux is the separation constant 70 var ( ' lambda_aux ') 71 KGradialpart = lambda_aux 72 KGangularpart = -lambda_aux 73 for term in fkgops : 74 if diff ( term , r ) ==0: 75 KGangularpart = KGangularpart + term 76 else : 77 KGradialpart = KGradialpart + term 78 79 KGradialpart = expand ( KGradialpart * R) . simplify_full () . collect ( R ) . collect ( diff (R , r ) ) . collect ( diff (R ,r , r ) ) 80 KGangularpart = expand ( KGangularpart * S ) . simplify_full () . collect ( S ) . collect ( diff (S , th ) ) . collect ( diff (S , th , th ) ) 81 110 111 # Substitute numerical values for parameters 112 radial2 = radial2 . subs ( M =0.1 , omega =0.2 , k =2.0 , lambda_aux =2.0) 113 13 114 # Solve the system and plot the solution 115 radsol = desolve_system_rk4 ([ radial1 , radial2 ] ,[R , R_aux ] , ics =[0.3 ,1 ,0.5] , ivar ...

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On numerical evaluation of the Heun functions

Cited by 18 publications

References 11 publications

On evaluation of the confluent Heun functions

On evaluation of the confluent Heun functions

Quantum field theory applications of heun type functions

Symbolic and numerical analysis in general relativity with open source computer algebra systems

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